3D Koch Snowflake

October 30th, 2008


Because of all the political advertisements, I find myself watching more and more PBS. I am a fan of NOVA, but this week’s topic was on fractals. Fractals have always struck me as boring; but compared to 30 smear campaigns they win every time.

A segment that caught my eye was about the Koch snowflake. It looks like a snowflake but has an interesting property in that it has a finite area but an infinite perimeter. Helge von Koch described this shape in a paper in 1904 –before he had a seat of SolidWorks, thus sadly his work was only in 2D.

I decided to continue the mathematician’s work in 3D. Instead of using triangles I would use tetrahedrons.

3D Koch Snowflake

3D Koch Snowflake eDrawing [Eight iterations; 4,373 tetrahedrons]

After I completed the model, I did a search for other 3D Koch Snowflakes. Turns out mine looks nothing like any others. You might be looking at the first, most accurate representation! Who are you going to believe -a guy who sat through an entire NOVA episode, or people who have devoted their entire lives to this “science”? <sigh> I guess maybe I should not say this is a 3D Koch Snowflake, but only that it was inspired by it. :(

I still think fractals are for hippies, but I thought this turned out pretty cool….and perhaps a fun challenge to you SolidWorks users out there, can you build it?

Edit: Here is the a six iteration in .stl format.3DKochSnowflake6Iterations

Jeff Sweeney

CSWE Engineering Data Specialist 3DVision Technologies

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14 Responses to “3D Koch Snowflake”

  1. Pete Yodis says:

    Jeff,

    Saw that PBS special too. It was fascinating. My favorite was perhaps the fractal relationship of branches to leaves on a tree correlating to the fractal relationship of small, medium, and large trees in the same forest. Second favorite is the use of fractal geometry in modern antennas. Got me thinking what fractal geometry I could cook up for a product. An electric motor with a fractal geometry for rotor and stator tooth geometry to improve electromagnetic interaction between the two???

  2. Jeff Sweeney says:

    I suppose they are both EM waves. You may be on to something! Remember me when you become famous with this idea!

  3. Seth Bischoff says:

    Amazing! How long did it take to create this? I want to see the model.

    On a side note, you should see some of the models Jeff creates after watching other programs!

  4. Jeff Sweeney says:

    There is a link to the eDrawing in this blog. I wanted to post the actual file, but it is over 99 MB…besides if I post it, it would be too easy to figure out how I made it! Seriously, if anyone wants the file let me know and I can put it on an ftp site for you.

    I am rather good at using the DOME feature.

  5. Tylor says:

    I’m in a more advanced geometry class right now, and I want to say that your representation is most definitely more accurate. The problem I see with the other representations (usually made in origami) is that they fail to see that the new added tetrahedrons have to be smaller. I’m really happy that you made this representation, because it gives me a better view on how the three-dimensional version of the Koch Snowflake works.

    I do have one suggestion though: on the original two-dimensional version, during the third iteration (first being the original triangle, and second being smaller triangles being added to it for the first time), shouldn’t there be more tetrahedrons added to the existing faces? Other than that, I’d have to completely agree with your model.

    Thanks a lot!

  6. Jeff Sweeney says:

    Thanks Taylor. I am not sure I under your question, the logic was that each face has one tetrahedron, adding more tetrahedrons would change the pattern wouldn’t it?

    I made each tetrahedron about 60% the size of the previous one. Now in retrospect, I think I should have made the size of the tetrahedrons shrink a little slower. Perhaps 65%, the size differential between the first three iterations seems a bit to much. ??

    Take the eDrawing version of this file in for your final. -Send me a copy of the A you get on your grade slip!

  7. Pete says:

    Jeff

    I believe I just saw the same PBS-NOVA special and came to a similar conclusion. In your model here you have placd your iterated solids centered on their parents’ triangular surface. The actual edges (or lines in the Koch Snowflake) are not being replaced by the motif. So, first you must decide what the 3D motif is. In 2D, it is 4 lines that are 1/3 the length of the original line. How I envision this in 3D is by placing a new tetrahedron which is 1/3 in size on each edge such that only one of its edges touch the original. I would love to see what you end up with using this pattern. Any SolidWorks takers?

  8. jenaro says:

    Hello Jeff,
    I am amazed by your 3D Koch Snowflake, I have been trying to replicate it in Studio Max but have failed to get the thing done. Is there any way you could handle a Solidworks stereolitho .stl file out? I would love to include an edited version of your snowflake in a piece of art i am doing :)
    best,

  9. Jeff Sweeney says:

    I have added a link to a six iteration version in .stl format in the main entry above.

  10. jenaro says:

    dear Jeff,
    thank you for adding the .stl file!
    I am working on it now, and I will be showing you the results later.
    best

  11. jenaro says:

    Dear Jeff,
    Thank you for collaborating! Here is the final result as to how the Koch snowflake got implemented into my art work.
    Thanx again, cheers :)

    http://www.behance.net/gallery/U-are-a-MIRROR-of-the-UNIVERSE/3504103

  12. Peter says:

    It’s certainly interesting, but I don’t think it’s a true fractal as some of the surface is still flat. All flat triangular surfaces should have a new tetrahedron protruding from them with each additional iteration. It should have a surface of infinite area bounding a finite volume
    http://cubicloud.blogspot.co.uk/2008/04/koch-snowflake-3d02.html shows the 2nd and 3rd iterations.
    Unfortunately, the tetrahedron seems unsuitable for making the ‘classic’ fractal look, as the extra shapes added with each iteration end up touching each other. I haven’t been able to find anyone make an image online like this that goes further than a handful of iterations!

  13. Jason Jiang says:

    Request permission to use an image

    Hi Mr. Sweeney,

    We are writing an essay on snowflake and plan to use one of your images on this page.

    The essay is written in Chinese and will be included into a book which will be published by “Beijing Normal University Publishing Group”. There is no plan to publish the essay on the internet. The size of the image is not determined but we expected it to be relatively small.

    We would like to ask you to allow us to use it. You will receive full credit.

    Sincerely,
    Jason

  14. Jeff Sweeney says:

    You may use the image Jason.

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